Home Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
Article Open Access

Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients

  • Sheng Zhang EMAIL logo and Luyao Zhang
Published/Copyright: March 7, 2016
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 14 Issue 1

Abstract

In this paper, Hirota’s bilinear method is extended to a new modified Kortweg–de Vries (mKdV) hierarchy with time-dependent coefficients. To begin with, we give a bilinear form of the mKdV hierarchy. Based on the bilinear form, we then obtain one-soliton, two-soliton and three-soliton solutions of the mKdV hierarchy. Finally, a uniform formula for the explicit N-soliton solution of the mKdV hierarchy is summarized. It is graphically shown that the obtained soliton solutions with time-dependent functions possess time-varying velocities in the process of propagation.

Keywords: Hirota’s bilinear method; bilinear form; soliton solution; mKdV hierarchy with time-dependent coefficients
PACS: 05.45.Yv; 02.30.Jr; 04.20.Jb

1 Introduction

In 1965, the concept of a soliton was first defined by Zabusky and Kruskal [1]. With the development of soliton theory, Zabusky and Kruskal’s classical solitons have been generalized to a more general class of solitons with at least time-varying velocities, such as those observed in nonlinear optics and Bose-Einstein condensate. It was Serkin, Hasegawa and Belyaeva [2] who substantially extended the concept of classical solitons to nonautonomous solitons. From a mathematical point of view, solitons are a class of special solutions of nonlinear partial differential equations (PDEs). As the generalization of classical differential equations with integer order, fractional-order differential calculus and its applications have attracted much attention [3–7]. In 2010, Fujioka, Espinosa and Rodríguez [8] described soliton propagation of an extended nonlinear Schrödinger equation which incorporates fractional dispersion and a fractional nonlinearity. Recently, Yang et al. [9] modeled fractal waves on shallow water surfaces via a local fractional KdV equation. In the past several decades, finding soliton solutions of nonlinear PDEs has gradually developed into a significant direction and some effective methods have been proposed for constructing soliton solutions, such as the inverse scattering method [10], Bäcklund transformation [11], Darboux transformation [12], Hirota’s bilinear method [13], Wronskian technique [14], tanh method [15], homogeneous balance method [16], Jacobi elliptic function expansion method [17], sub-equation method [18], exp-function method [19], transformed rational function method [20], and the first integral method [21, 22].

As a direct method, Hirota’s bilinear method [13] has been widely used to construct multi-soliton solutions of many nonlinear PDEs [23–34]. However, there is very little research work in extending Hirota’s bilinear method for a whole hierarchy of nonlinear PDEs (see. e.g., [35]). This is because it is difficult to find a bilinear form suitable for all the nonlinear PDEs in a given hierarchy. In this paper, we shall give a bilinear form of the following new mKdV hierarchy with time-dependent coefficients

(1)vt = ∑m = 0nα2m+1(t)Fmvx, n = 0,1, 2,⋅⋅⋅,

and then construct its multi-soliton solutions through Hirota’s bilinear method. Here the recursion operator employed is

F = ∂2 + 4v2 + 4vx∂−1v,∂ = ∂∂x, ∂−1 = 12  (∫−∞xdx − ∫x∞dx).

When α2m+1(t) = 1 and m =n, Equation (1) becomes the known constant-coefficient mKdV hierarchy [30] vt = FnVx, n = 0, 1, 2, ···. The first equation of the mKdV hierarchy (1) is the trivial linear variable-coefficient equation vt = α1(t)vx, and the second and third equations are the following non-trivial nonlinear equations with variable-coefficients:

(2)vt = α1(t)vx +α3(t)vxxx+6α3(t)v2vx,
(3)vt = α1(t)vx + α3(t)vxxx + 6α3(t)v2vx  +α5(t)vxxxxx + 10α5(t)v2vxxx  +40α5(t)vvxvxx + 10α5(t)vx3 + 30α5(t)v4vx.

Obviously, Equations (2) and (3) include the famous mKdV equation vt = vxxx + 6v2vx as a special case.

In Section 2, a bilinear form of the mKdV hierarchy (1) is obtained through Hirota’s bilinear method. In Section 3, starting from the obtained bilinear form, we first construct one-soliton, two-soliton, and three-soliton solutions of the mKdV hierarchy (1). Based on the obtained soliton solutions, we then summarize a uniform formula for the explicit N-soliton solution of the mKdV hierarchy (1). In addition, some spatial structures and propagations of the obtained one-, two- and three-soliton solutions are shown in figures. In Section 4, we conclude this paper.

2 Bilinearization

For the bilinear form of the mKdV hierarchy (1), we have the following theorem.

Theorem 1.

The mKdV hierarchy (1)possesses a bilinear form:

(4)Dx2f* • f =  0,
(5)( Dt − ∑m=0nα2m+1(t)Dx2m+1)  f* ⋅ f = 0,
where f*denotes the conjugate of f =f (x,t ), and Dx and Dtare Hirota’s differential operators defined by
DxmDtnf(x,t)·g(x,t)=(∂x−∂x')m(∂t−∂t')nf(x,t)·g(x',t')|x'=x,t'=t·

Now, we introduce the following theorem:

Proof

Firstly, we reduce the general term α2m+1(t)Fm vx in Equation (1). For the sake of convenience, we introduce

(6)vt2m+1 = α2m+1(t)Fmvx,  m  =  0,1,2,⋯, n.

For any integer m ≥ 1, Equation (6) can be rewritten as

(7)vt2m+1 = α2m+1(t)α2m−1(t)Fvt2m−1 = α2m+1(t)α2m−1(t)[vt2m−1xx + 4v2vt2m−1 +  4vx∂−1vvt2m−1].

Integrating Equation (7) with respect to x once, we have

(8)∂−1Vt2m+1 =  α2m+1(t)α2m−1(t) [vt2m−1x+2v∂−1(v2)t2m−1]  .

Secondly, we take a logarithmic transformation

(9)v = i(lnf*f)x, f = f(x,t1,t3,⋯),

where i is the imaginary unit. Then Equation (8) becomes

(10)1f*fDt2m+1f*  ⋅ f           =α2m+1(t)α2m−1(t)[1f*fDx2Dt2m−1f*  ⋅ f           −1f*2f2(Dt2m−1f*⋅ f)(Dx2f* ⋅ f)           −2f*f(Dxf* ⋅ f)∂−1(Dx2f* ⋅ ff*f)t2m−1].

If we set Dx2f* · f =  0,, then Equation (10) is reduced to

(11)Dt2m+1f*  ⋅ f = α2m+1(t)α2m−1(t)Dx2Dt2m−1f*  ⋅  f.

From Equations (6), (7) and (11), we have

(12)Dt2m+1f*  ⋅  f  = α2m+1(t)Dx2m+1f*  ⋅ f,  m  =  0,1,2, ⋯,n.

It is convenient to introduce t1, t2, ···, tn for the iterative reduction of the general term α2m+1(t)Fmvx in Equation (6). For example, Equation (6) gives

(13)vt1 = α1(t)vx, vt3 = α3(t)Fvx = α3(t)α1(t)FVt1,
(14)vt5 =  α5(t)F2vx  =  α5(t)α3(t)Fvt3,

and so forth.

In fact,t1, t2, · · ·, tn can be seen as the same as t. It is then easy to see from Equations (11)–(14) that the right side of Equation (1) is expressed by

(15)∑m=0nα2m+1(t)FmVx = 1f*f ∑m=0nα2m+1(t)Dx2m+1f*  ⋅ f

under the case of Dx2f* · f =  0,. At the same time, for the left side of Equation (1) we have

(16)vt  =  1f*fDtf*  ⋅ f. 

Finally, with the help of Equations (14) and (15) we obtain a bilinear form which consists of Equations (4) and (5) for the mKdV hierarchy (1). Thus, the proof is complete. □

3 Multi-soliton solutions

In this section, we use the bilinear form in Equations (4) and (5) to construct multi-soliton solutions of the mKdV hierarchy (1). In order to obtain the one-soliton solution, we suppose that

(17)f   =  1+εf(1)+ε2f(2)+ε3f(3)+⋯,
(18)f*   =  1+εf(1)*+ε2f(2)* +ε3f(3)*+⋯ . 

Substituting Equations (17) and (18) into Equations (4) and (5), and then collecting the coefficients of the same order of 𝜀, we obtain a system of PDEs:

(19)∂x2(f(1)+f(1)*)  = 0,
(20)∂x2(f(2)+f(2)*)  =  −Dx2f(1) ⋅  f(1)*,
(21)∂x2(f(3)+f(3)*)  = −Dx2(f(1)  ⋅  f(2)* + f(1)* ⋅  f(2)),
(22)[∂t − ∑m=0nα2m+1(t)∂x2m+1] (f(1) + f(1)*)  = 0,
(23)[∂t − ∑m=0nα2m+1(t)∂x2m+1] (f(2) + f(2)*)=−(Dt − ∑m=0nα2m+1(t)Dx2m+1)f(1)  ⋅ f(1)*,
(24)[∂t−∑m=0nα2m+1(t)∂x2m+1](f(3)+f(3)*)=−(Dt−∑m=0nα2m+1(t)Dx2m+1)×(f(1)⋅f(2)*+f(1)*⋅f(2)),⋮

and so forth. If we let

(25)f(1) = eξ1+π2i, ξ1 = k1x+∑​​m=0nw1,2m+1  ∫​​​α2m+1(t)dt

be a solution of Equations (19) and (22), we obtain

(26)w1,2m+1 = k12m+1.

Substituting Equations (25) and (26) into Equations (20) and (23), we can verify that if

(27)f(2) = f(2)∗ = 0,

then Equations (21), (24) and all the other equations omitted in the above system of PDEs all hold. In this case, we write

(28)f1 = 1 +  eξ1 + π2i,

and hence obtain the following one-soliton solution of the mKdV hierarchy (1):

(29)v = i(lnf1∗f1)x = k1 sechξ1.

In Figure 1, the spatial structure of the one-soliton solution (29) is shown. Figure 2 displays the propagation of the one-soliton solution (29) along the negative x-axis. It can be seen from the velocity image in Figure 3 that the velocity of solution (29) periodically changes over time in the process of propagation.

Figure 1. Spatial structure of one-soliton solution (29) with k1 = 0.7, 2m+1(t) = 1 + 0.5 m sin t, n = 20.
Figure 1.

Spatial structure of one-soliton solution (29) with k1 = 0.7, 2m+1(t) = 1 + 0.5 m sin t, n = 20.

Figure 2. Propagation of one-soliton solution (29) with k1 = 0.7, α2m + 1(t) = 1 + 0.5 m sin t, n = 20 at different times (a) t = −5, (b) t = 0 and (c) t = 5.
Figure 2.

Propagation of one-soliton solution (29) with k1 = 0.7, α2m + 1(t) = 1 + 0.5 m sin t, n = 20 at different times (a) t = −5, (b) t = 0 and (c) t = 5.

Figure 3. Velocity image of one-soliton solution (29) with k1 = 0.7, α2m + 1(t) = 1 + 0.5 m sin t, n = 20.
Figure 3.

Velocity image of one-soliton solution (29) with k1 = 0.7, α2m + 1(t) = 1 + 0.5 m sin t, n = 20.

For the two-soliton solution, we suppose that

(30)f(1) = eξ1 + π2i + eξ2 + π2i, 

with

ξ2 = k2x + ∑​​m=0nw2,2m+1∫​​​α2m+1(t)dt.

Substituting Equation (30) into Equations (19) and (22) yields

(31)w2,2m+1 = k22m+1.

In view of Equation (23), we further suppose that

(32)f(2) = eξ1+ξ2+πi+θ12,

where θ12 is a constant to be determined. Substituting Equations (30)–(32) into Equation (20), we obtain

(33)eθ12 = (k1 − k2)2(k1 − k2)2.

It is easy to see that if we substitute Equations (30)–(32) into Equations (21) and (24), then we can verify that

(34)f(3) = f(3)* = ⋯ = 0

satisfies Equations (21), (24) and all the other equations omitted in the above system of PDEs. In this case, we write

(35)f2=1+eξ1+π2i+ eξ2+π2i + eξ1+ξ2+πi+θ12

and hence obtain the following two-soliton solution of the mKdV hierarchy (1):

(36)v=i[In(1+eξ1−π2i+ eξ2−π2i + eξ1+ξ2−πi+θ121+eξ1+π2i+ eξ2+π2i + eξ1+ξ2+πi+θ12)]x.

In Figure 4, the spatial structure of the two-soliton solution (36) is shown. Figure 5 displays the variable velocity propagation of the two-soliton solution (36) along the negative x-axis.

Figure 4. Spatial structure of two-soliton solution (36) with k1 = 0.3, k2 = 0.6, α2m + 1(t) = 1 + 2m sin t, n = 20.
Figure 4.

Spatial structure of two-soliton solution (36) with k1 = 0.3, k2 = 0.6, α2m + 1(t) = 1 + 2m sin t, n = 20.

Figure 5. Propagation of two-soliton solution (36) with k1 = 0.3, k2 = 0.6, α2m + 1(t) = 1 + 2m sin t, n = 120 at different times (a) t = −10, (b) t = 0 and (c) t = 10.
Figure 5.

Propagation of two-soliton solution (36) with k1 = 0.3, k2 = 0.6, α2m + 1(t) = 1 + 2m sin t, n = 120 at different times (a) t = −10, (b) t = 0 and (c) t = 10.

Similarly, we continue to construct the three-soliton solution. For this purpose, we suppose that

(37)f(1) = eξ1+π2i+ eξ2+π2i + eξ3+π2i,

with

ξ3 = k3x + ∑m=0nw3,2m+1∫​α2m+1(t)dt.

Substituting Equation (37) into Equations (19) and (22) yields

(38)w3,2m+1 = k32m+1. 

In view of Equations (20) and (23), we suppose that

(39)f(2) = eξ1+ξ2+πi+θ12+ eξ1+ξ3+πi+θ13+ eξ2+ξ3+πi+θ23,

where θ13 and θ23 are unknown constants to be determined.

Substituting Equations (37)–(39) into Equations (20) and (23), we obtain

(40)eθ13 = (k1 − k3)2(k1 − k3)2, eθ23 = (k2 − k3)2(k2 − k3)2. 

Then the substitution of Equations (37)–(40) into Equations (21) and (24) gives

f(3) = eξ1+π2i+ξ2+π2i+ξ3+π2i+θ12+θ13+θ23.

If we further take

(41)f(i) = 0,f(i)* = 0,i = 4,5,...,

then it can be verified that all the other equations omitted in the above system of PDEs hold. In this case, we write

(42)f3 = 1 +eξ1+π2i+eξ2+π2i+eξ3+π2i+eξ1+ξ2+πi+θ12+eξ1+ξ3+πi+θ13+eξ2+ξ3+πi+θ23+eξ1+ξ2+ξ3+θ12+θ13+θ23+32πi,

and hence obtain the following three-soliton solution of the the mKdV hierarchy (1):

(43)v = i[ln(1+eξ1-π2i+eξ2-π2i+eξ3−π2i+eξ1+ξ2−πi+θ12+eξ1+ξ3−πi+θ13+eξ2+ξ3−πi+θ23+eξ1+ξ2+ξ3+θ12+θ13+θ23−32πi1+eξ1+π2i+eξ2+π2i+eξ3+π2i+eξ1+ξ2+πi+θ12+eξ1+ξ3+πi+θ13+eξ2+ξ3+πi+θ23+eξ1+ξ2+ξ3+θ12+θ13+θ23+32πi)]x.

In Figure 6, the spatial structure of the three-soliton solution (43) is shown. Figure 7 displays the variable velocity propagation of the three-soliton solution (43) along the negative x-axis.

Figure 6. Spatial structure of three-soliton solution (43) with k1 = 0.5, k2 = 1, k3 = 1, α2m + 1(t) = (1+mt2)−1, n = 10.
Figure 6.

Spatial structure of three-soliton solution (43) with k1 = 0.5, k2 = 1, k3 = 1, α2m + 1(t) = (1+mt2)−1, n = 10.

Figure 7. Propagation of one-soliton solution (29) with k1 = 0.5, k2 = 1.05, k3 = 1, α2m + 1(t) = (1 + mt2)−1n = 10 at different times (a) t = −15, (b) t = 0 and (c) t = 15.
Figure 7.

Propagation of one-soliton solution (29) with k1 = 0.5, k2 = 1.05, k3 = 1, α2m + 1(t) = (1 + mt2)−1n = 10 at different times (a) t = −15, (b) t = 0 and (c) t = 15.

Generally, if we take

(44)fN = ∑μ=0,1e∑i=1Nμi(ξi+π2i)+∑i≤i≤jμiμjθij,
(45)ξi = ki x+∑m=0nwi,2m+1 ∫α2m+1(t)dt, wi,2m+1 = ki2m+1,
(46)eθij=(ki−kj)2(ki−kj)2(i<j;i,j=1,2,⋯,N),

then a uniform formula of the explicit N-soliton solution of the mKdV hierarchy (1) can be obtained as follows:

(47)v = i{In[∑μ=0,1e∑i=1Nμi(ξi−π2i)+∑1≤i≤jμiμjθij∑μ=0,1e∑i=1Nμi(ξi−π2i)+∑1≤i≤jμiμjθij]}x,

where the summation ∑μ-0, 1 refers to all possible combinations of each μi = 0, 1 for i = 1, 2,..., N.

To the best of our knowledge, the one-soliton solution (29), two-soliton solution (36), three-soliton solution (43) and N -soliton solution (47) are new; they have not been reported in literature.

4 Conclusions

In summary, we have extended Hirota’s bilinear method to the mKdV hierarchy (1) with time-dependent coefficients. In the procedure of extending Hirota’s bilinear method to the mKdV hierarchy (1), one of the key steps is to reduce this mKdV hierarchy (1) to the bilinear form in Equations (4) and (5). The obtained one-soliton, two-soliton, three-soliton and N -soliton solutions (29), (36), (43) and (47) contain many time-dependent functions which provide enough freedom for us to describe spatial structures and propagations of these solutions. It is graphically shown that the one-soliton, two-soliton and three-soliton solutions (29), (36) and (43) possess time-varying velocities in the process of propagation. How to extend Hirota’s bilinear method to some other hierarchies of nonlinear PDEs with time-dependent coefficients is worthy of study. This is our task in the future.

Acknowledgement

We would like to express our sincerest thanks to the referees for their valuable suggestions and comments. This work was supported by the PhD Startup Funds of Liaoning Province of China (20141137) and Bohai University (bsqd2013025), the Natural Science Foundation of Educational Committee of Liaoning Province of China (L2012404), the Liaoning BaiQianWan Talents Program (2013921055) and the Natural Science Foundation of China (11547005).

References

1 N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)10.1103/PhysRevLett.15.240Search in Google Scholar

2 V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007)10.1103/PhysRevLett.98.074102Search in Google Scholar

3 X.J. Yang, H.M. Srivastava, C. Cattani, Rom. Rep. Phys. 67, 752 (2015)Search in Google Scholar

4 X.J. Yang, H.M. Srivastava, Commun. Nonlinear Sci. 29, 499 (2015)10.1016/j.cnsns.2015.06.006Search in Google Scholar

5 X.J. Yang, D. Baleanu, H.M. Srivastava, Appl. Math. Lett. 47, 54 (2015)10.1016/j.aml.2015.02.024Search in Google Scholar

6 X.J. Yang, D. Baleanu, P. Lazarevi[cacute] Mihailo, S. Caji[cacute] Milan, Therm. Sci. 19, 959 (2015)Search in Google Scholar

7 X.J. Yang, D. Baleanu, H.M. Srivastava, Local Fractional Integral Transforms and Their Applications (Academic Press, New York, 2015)10.1016/B978-0-12-804002-7.00004-8Search in Google Scholar

8 J. Fujioka, A. Espinosa, R.F. Rodríguez, Phys. Lett. A, 374, 1126 (2010)10.1016/j.physleta.2009.12.051Search in Google Scholar

9 X.J. Yang et al., Abstr. Appl. Anal. 2014, 278672 (2014)Search in Google Scholar

10 C.S. Gardner et al., Phys. Rev. Lett. 19, 1095 (1967)10.1103/PhysRevLett.19.1095Search in Google Scholar

11 M.R. Miurs, Bäcklund Transformation (Springer, Berlin, 1978)Search in Google Scholar

12 X.Y. Wen, Rep. Math. Phys. 68, 211 (2011)10.1016/S0034-4877(12)60005-6Search in Google Scholar

13 R. Hirota, Phys. Rev. Lett. 27, 1192, (1971)10.1103/PhysRevLett.27.1192Search in Google Scholar

14 N.C. Freeman, J. J. C. Nimmo, Phys. Lett. A 95, 1 (1983)10.1016/0375-9601(83)90764-8Search in Google Scholar

15 W. Malfliet, Amer. J. Phys. 60, 65 (1992)10.1016/0006-3207(92)90821-4Search in Google Scholar

16 M.L. Wang, Phys. Lett. A 213, 279 (1996)10.1016/0375-9601(96)00103-XSearch in Google Scholar

17 S.K. Liu et al., Phys. Lett. A 289, 69 (2001)10.1016/S0378-3820(01)00158-8Search in Google Scholar

18 E.G. Fan, Phys. Lett. A 300, 243 (2002)10.1016/S0375-9601(02)00776-4Search in Google Scholar

19 J.H. He, X.H. Wu, Chaos Soliton. Fract. 30, 700 (2006)10.1016/j.chaos.2006.03.020Search in Google Scholar

20 W.X. Ma, J.H. Lee, Chaos Soliton. Fract. 42, 1356 (2009)10.1016/j.chaos.2009.03.043Search in Google Scholar

21 D. Baleanu, B. Kilic, I. Mustafa, P. Romanian Acad. A 16, 160 (2015)Search in Google Scholar

22 D. Baleanu et al. Rom. J. Phys. 60, 111 (2015)Search in Google Scholar

23 R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, New York, 2004)10.1017/CBO9780511543043Search in Google Scholar

24 M.V. Balashov, Math. Notes 71, 339 (2005)10.1155/IMRN.2005.339Search in Google Scholar

25 R. Hirota, Prog. Theor. Phys. 52, 1498 (1974)10.1143/PTP.52.1498Search in Google Scholar

26 R. Hirota, J. Satsuma, Prog. Theor. Phys. Supp. 59, 64 (1976)10.1143/PTPS.59.64Search in Google Scholar

27 R. Hirota, J. Phys. Soc. Jpn. 33, 1456 (1972)10.1143/JPSJ.33.1456Search in Google Scholar

28 I. Mcarthur, C.M. Yung, Mod. Phys. Lett. A 8, 1739 (1993)10.1142/S0217732393001471Search in Google Scholar

29 Q.P. Liu, X.B. Hu, M.X. Zhang, Nonlinearity, 18, 1597 (2005)10.1088/0951-7715/18/4/009Search in Google Scholar

30 D.Y. Chen, Introduction to Soliton (Science press, Beijing, 2006) (in Chinese)Search in Google Scholar

31 A. Arikoglu, I. Ozkol, Appl. Math. Comput. 181, 153 (2006)10.1016/j.amc.2006.01.022Search in Google Scholar

32 A.M. Wazwaz, Appl. Math. Comput. 200, 160 (2008)10.1016/j.amc.2007.11.001Search in Google Scholar

33 E.G. Fan, Y.C. Hon, Stud. Appl. Math. 125, 343 (2010)10.1111/j.1467-9590.2010.00491.xSearch in Google Scholar

34 E.G. Fan, Y.C. Chow, Stud. Appl. Math. 127, 284 (2012)10.1111/j.1467-9590.2011.00520.xSearch in Google Scholar

35 D.Y. Chen, et al., Chin. Ann. Math. Ser. A 33, 206 (2012) (in Chinese)10.1007/s11401-012-0724-1Search in Google Scholar

Received: 2015-10-23
Accepted: 2015-12-7
Published Online: 2016-3-7
Published in Print: 2016-1-1

© 2016 S. Zhang and L. Zhang published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular articles
  2. Speeding of α Decay in Strong Laser Fields
  3. Regular articles
  4. Multi-soliton rational solutions for some nonlinear evolution equations
  5. Regular articles
  6. Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
  7. Regular articles
  8. Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
  9. Regular articles
  10. Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
  11. Regular articles
  12. Modeling the potential energy field caused by mass density distribution with Eton approach
  13. Regular articles
  14. Climate Solutions based on advanced scientific discoveries of Allatra physics
  15. Regular articles
  16. Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
  17. Regular articles
  18. Synthesis of Pt nanowires with the participation of physical vapour deposition
  19. Regular articles
  20. Quantum discord and entanglement in grover search algorithm
  21. Regular articles
  22. On order statistics from nonidentical discrete random variables
  23. Regular articles
  24. Charmed hadron photoproduction at COMPASS
  25. Regular articles
  26. Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
  27. Regular articles
  28. Flap motion of helicopter rotors with novel, dynamic stall model
  29. Regular articles
  30. Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
  31. Regular articles
  32. Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
  33. Regular articles
  34. Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
  35. Regular articles
  36. Recent Progress in Search for Dark Sector Signatures
  37. Regular articles
  38. Recent progress in organic spintronics
  39. Regular articles
  40. On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
  41. Regular articles
  42. Self-healing phenomena of graphene: potential and applications
  43. Regular articles
  44. Viscous flow and heat transfer over an unsteady stretching surface
  45. Regular articles
  46. Spacetime Exterior to a Star: Against Asymptotic Flatness
  47. Regular articles
  48. Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
  49. Regular articles
  50. Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
  51. Regular articles
  52. Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
  53. Regular articles
  54. Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
  55. Regular articles
  56. Unification of all hyperbolic tangent function methods
  57. Regular articles
  58. Analytical solution for the correlator with Gribov propagators
  59. Regular articles
  60. A New Algorithm for the Approximation of the Schrödinger Equation
  61. Regular articles
  62. Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
  63. Regular articles
  64. On the fractional differential equations with not instantaneous impulses
  65. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  66. Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
  67. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  68. Numerical solution of two dimensional time fractional-order biological population model
  69. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  70. Rotational surfaces in isotropic spaces satisfying weingarten conditions
  71. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  72. Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
  73. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  74. Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
  75. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  76. Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
  77. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  78. Logical entropy of quantum dynamical systems
  79. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  80. An efficient algorithm for solving fractional differential equations with boundary conditions
  81. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  82. A numerical method for solving systems of higher order linear functional differential equations
  83. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  84. Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
  85. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  86. On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
  87. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  88. On soliton solutions of the Wu-Zhang system
  89. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  90. Comparison between the (G’/G) - expansion method and the modified extended tanh method
  91. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  92. On the union of graded prime ideals
  93. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  94. Oscillation criteria for nonlinear fractional differential equation with damping term
  95. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  96. A new method for computing the reliability of consecutive k-out-of-n:F systems
  97. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  98. A time-delay equation: well-posedness to optimal control
  99. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  100. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
  101. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  102. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
  103. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  104. The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
  105. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  106. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
  107. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  108. A certain sequence of functions involving the Aleph function
  109. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  110. On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
  111. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  112. Solitary and compacton solutions of fractional KdV-like equations
  113. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  114. Regarding on the exact solutions for the nonlinear fractional differential equations
  115. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  116. Non-local Integrals and Derivatives on Fractal Sets with Applications
  117. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  118. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
  119. Special issue on Information Technology and Computational Physics
  120. On uninorms and nullnorms on direct product of bounded lattices
  121. Special issue on Information Technology and Computational Physics
  122. Phase-space description of the coherent state dynamics in a small one-dimensional system
  123. Special issue on Information Technology and Computational Physics
  124. Automated Program Design – an Example Solving a Weather Forecasting Problem
  125. Special issue on Information Technology and Computational Physics
  126. Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
  127. Special issue on Information Technology and Computational Physics
  128. Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
  129. Special issue on Information Technology and Computational Physics
  130. Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
  131. Special issue on Information Technology and Computational Physics
  132. Intelligent Chatter Bot for Regulation Search
  133. Special issue on Information Technology and Computational Physics
  134. Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
  135. Special issue on Information Technology and Computational Physics
  136. Resource management for server virtualization under the limitations of recovery time objective
  137. Special issue on Information Technology and Computational Physics
  138. MODY – calculation of ordered structures by symmetry-adapted functions
  139. Special issue on Information Technology and Computational Physics
  140. Survey of Object-Based Data Reduction Techniques in Observational Astronomy
  141. Special issue on Information Technology and Computational Physics
  142. Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
  143. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  144. Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
  145. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  150. Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
  151. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  152. Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
  153. Special Issue on Research Frontier on Molecular Reaction Dynamics
  154. Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
  155. Special Issue on Research Frontier on Molecular Reaction Dynamics
  156. Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
  157. Special Issue on Research Frontier on Molecular Reaction Dynamics
  158. The Interaction between Graphene and Oxygen Atom
  159. Special Issue on Research Frontier on Molecular Reaction Dynamics
  160. Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
  161. Special Issue: Functional Advanced and Nanomaterials
  162. Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
  163. Special Issue: Functional Advanced and Nanomaterials
  164. Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
  165. Special Issue: Functional Advanced and Nanomaterials
  166. Quasi-planar elemental clusters in pair interactions approximation
  167. Special Issue: Functional Advanced and Nanomaterials
  168. Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate
https://doi.org/10.1515/phys-2016-0002
Keywords for this article
Hirota’s bilinear method; bilinear form; soliton solution; mKdV hierarchy with time-dependent coefficients
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular articles
  2. Speeding of α Decay in Strong Laser Fields
  3. Regular articles
  4. Multi-soliton rational solutions for some nonlinear evolution equations
  5. Regular articles
  6. Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
  7. Regular articles
  8. Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
  9. Regular articles
  10. Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
  11. Regular articles
  12. Modeling the potential energy field caused by mass density distribution with Eton approach
  13. Regular articles
  14. Climate Solutions based on advanced scientific discoveries of Allatra physics
  15. Regular articles
  16. Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
  17. Regular articles
  18. Synthesis of Pt nanowires with the participation of physical vapour deposition
  19. Regular articles
  20. Quantum discord and entanglement in grover search algorithm
  21. Regular articles
  22. On order statistics from nonidentical discrete random variables
  23. Regular articles
  24. Charmed hadron photoproduction at COMPASS
  25. Regular articles
  26. Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
  27. Regular articles
  28. Flap motion of helicopter rotors with novel, dynamic stall model
  29. Regular articles
  30. Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
  31. Regular articles
  32. Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
  33. Regular articles
  34. Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
  35. Regular articles
  36. Recent Progress in Search for Dark Sector Signatures
  37. Regular articles
  38. Recent progress in organic spintronics
  39. Regular articles
  40. On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
  41. Regular articles
  42. Self-healing phenomena of graphene: potential and applications
  43. Regular articles
  44. Viscous flow and heat transfer over an unsteady stretching surface
  45. Regular articles
  46. Spacetime Exterior to a Star: Against Asymptotic Flatness
  47. Regular articles
  48. Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
  49. Regular articles
  50. Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
  51. Regular articles
  52. Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
  53. Regular articles
  54. Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
  55. Regular articles
  56. Unification of all hyperbolic tangent function methods
  57. Regular articles
  58. Analytical solution for the correlator with Gribov propagators
  59. Regular articles
  60. A New Algorithm for the Approximation of the Schrödinger Equation
  61. Regular articles
  62. Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
  63. Regular articles
  64. On the fractional differential equations with not instantaneous impulses
  65. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  66. Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
  67. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  68. Numerical solution of two dimensional time fractional-order biological population model
  69. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  70. Rotational surfaces in isotropic spaces satisfying weingarten conditions
  71. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  72. Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
  73. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  74. Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
  75. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  76. Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
  77. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  78. Logical entropy of quantum dynamical systems
  79. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  80. An efficient algorithm for solving fractional differential equations with boundary conditions
  81. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  82. A numerical method for solving systems of higher order linear functional differential equations
  83. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  84. Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
  85. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  86. On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
  87. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  88. On soliton solutions of the Wu-Zhang system
  89. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  90. Comparison between the (G’/G) - expansion method and the modified extended tanh method
  91. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  92. On the union of graded prime ideals
  93. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  94. Oscillation criteria for nonlinear fractional differential equation with damping term
  95. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  96. A new method for computing the reliability of consecutive k-out-of-n:F systems
  97. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  98. A time-delay equation: well-posedness to optimal control
  99. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  100. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
  101. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  102. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
  103. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  104. The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
  105. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  106. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
  107. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  108. A certain sequence of functions involving the Aleph function
  109. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  110. On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
  111. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  112. Solitary and compacton solutions of fractional KdV-like equations
  113. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  114. Regarding on the exact solutions for the nonlinear fractional differential equations
  115. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  116. Non-local Integrals and Derivatives on Fractal Sets with Applications
  117. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  118. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
  119. Special issue on Information Technology and Computational Physics
  120. On uninorms and nullnorms on direct product of bounded lattices
  121. Special issue on Information Technology and Computational Physics
  122. Phase-space description of the coherent state dynamics in a small one-dimensional system
  123. Special issue on Information Technology and Computational Physics
  124. Automated Program Design – an Example Solving a Weather Forecasting Problem
  125. Special issue on Information Technology and Computational Physics
  126. Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
  127. Special issue on Information Technology and Computational Physics
  128. Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
  129. Special issue on Information Technology and Computational Physics
  130. Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
  131. Special issue on Information Technology and Computational Physics
  132. Intelligent Chatter Bot for Regulation Search
  133. Special issue on Information Technology and Computational Physics
  134. Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
  135. Special issue on Information Technology and Computational Physics
  136. Resource management for server virtualization under the limitations of recovery time objective
  137. Special issue on Information Technology and Computational Physics
  138. MODY – calculation of ordered structures by symmetry-adapted functions
  139. Special issue on Information Technology and Computational Physics
  140. Survey of Object-Based Data Reduction Techniques in Observational Astronomy
  141. Special issue on Information Technology and Computational Physics
  142. Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
  143. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  144. Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
  145. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  150. Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
  151. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  152. Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
  153. Special Issue on Research Frontier on Molecular Reaction Dynamics
  154. Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
  155. Special Issue on Research Frontier on Molecular Reaction Dynamics
  156. Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
  157. Special Issue on Research Frontier on Molecular Reaction Dynamics
  158. The Interaction between Graphene and Oxygen Atom
  159. Special Issue on Research Frontier on Molecular Reaction Dynamics
  160. Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
  161. Special Issue: Functional Advanced and Nanomaterials
  162. Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
  163. Special Issue: Functional Advanced and Nanomaterials
  164. Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
  165. Special Issue: Functional Advanced and Nanomaterials
  166. Quasi-planar elemental clusters in pair interactions approximation
  167. Special Issue: Functional Advanced and Nanomaterials
  168. Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 17.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2016-0002/html
Scroll to top button